Nowhere-zero $3$-flow of graphs with small independence number

TL;DR

This paper characterizes graphs with small independence number that admit a nowhere-zero 3-flow, verifying Tutte's conjecture for these graphs and introducing new reduction techniques for odd wheels.

Contribution

It provides a complete characterization of graphs with independence number at most 4 that admit a nowhere-zero 3-flow and proves the conjecture for graphs with independence number at most 3 and certain connectivity.

Findings

Characterization of graphs with independence number ≤4 admitting a nowhere-zero 3-flow

Verification of Tutte's 3-flow conjecture for graphs with independence number ≤4 and order ≥21

Proof that odd-5-edge-connected graphs with independence number ≤3 admit a nowhere-zero 3-flow

Abstract

Tutte's $3$-flow conjecture states that every $4$-edge-connected graph admits a nowhere-zero $3$-flow. In this paper, we characterize all graphs with independence number at most $4$ that admit a nowhere-zero $3$-flow. The characterization of $3$-flow verifies Tutte's $3$-flow conjecture for graphs with independence number at most $4$ and with order at least $21$. In addition, we prove that every odd-$5$-edge-connected graph with independence number at most $3$ admits a nowhere-zero $3$-flow. To obtain these results, we introduce a new reduction method to handle odd wheels.